No real problems at all.
One serieus thing, proof of insurance. The LBA will not register an aircraft to your name if you do not send it together with the change of owner ship form. And I really mean together. Do not accept that your insurer send it for you, it will never arrive on the right LBA desk.
Better, go in person, with the former owner, to the LBA in Braunsweig.
The form for change of owner ship is somewhere on the LBA website(also in English)
Check if there are no liens on that aircraft.
Check if the owner is really the owner.
If the plane has an ELT and/or mode-s, change the registration(s) to your name.
And you have to get a license for the radios and nav. stuff.
There was a web-site once that covered all these things and some more but I've lost the link.
-Kees
I've found that link:
What is involved when buying an aircraft (in Germany)